作図と対称性 - Speaker Deck

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2017೥10݄9೔ ࡞ਤͱରশੑ ക࡚௚໵

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ࠓ͔Β221 = 13 × 17೥લ Gaussਖ਼17֯ܗͷ࡞ਤ๏Λൃݟ 2017೥10݄7೔ mathpower2017

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ਖ਼65537֯ܗͷ࡞ਤ

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65537͸ʁ

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ૉ਺

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ϑΣϧϚʔૉ਺ 3 = 2 + 1 5 = 4 + 1 = 2 × 2 + 1 17 = 16 + 1 = 4 × 4 + 1 257 = 256 + 1 = 16 × 16 + 1 65537 = 65536 + 1 = 256 × 256 + 1

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ͪͳΈʹ 4294967297 = 65536 × 65536 + 1 = 641 × 6700417 18446744073709551617 = 4294967296 × 4294967296 + 1 = 274177 × 67280421310721 · · · 65537͸ʢࠓͷͱ͜Ζʣ࠷େͷϑΣϧϚʔૉ਺

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͍ͭͰ΋ ࡞ਤͰ͖Δʁ

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ਖ਼3֯ܗͷ࡞ਤ

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120◦

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120◦ 60◦

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60◦ 60◦ 60◦ 1 1 2

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ίϯύεͱఆنΛ࢖͑͹ɺ௕͞Λೋ౳෼Ͱ͖Δɻ A B

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௕͞1ͷઢ෼͔Β ௕͞ 1 2 ͷઢ෼Λ࡞Δ͜ͱ͕Ͱ͖ͨʂ

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࡞ਤͰͰ͖Δ͜ͱ • ਨ௚ͳઢΛҾ͘ • ฏߦͳઢΛҾ͘ • ௕͞Λೋ౳෼͢Δ

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ਖ਼3֯ܗͷ࡞ਤ

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࡞ਤΛ͢Δ=৽͍͠௕͞Λ࡞Δ=৽͍͠਺Λ࡞Δ ௕͞1ͷઢ෼AB ͕༩͑ΒΕ͍ͯΔͱ͢Δɻ

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࡞ਤͰͰ͖Δ͜ͱ • ਨ௚ͳઢΛҾ͘ • ฏߦͳઢΛҾ͘ • ௕͞Λೋ౳෼͢Δ • ௕͞ͷ଍͠ࢉͱҾ͖ࢉΛ͢Δ

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௕͞ 4 3 ͷ࡞Γํ 3 4

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࡞ਤͰͰ͖Δ͜ͱ • ਨ௚ͳઢΛҾ͘ • ฏߦͳઢΛҾ͘ • ௕͞Λೋ౳෼͢Δ • ௕͞ͷ଍͠ࢉͱҾ͖ࢉΛ͢Δ • ௕͞ͷֻ͚ࢉͱׂΓࢉΛ͢Δ

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࣮͸ √ 3΋࡞ਤ͍ͯͨ͠

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ϐλΰϥεͷఆཧ x2 + y2 = r2 √ 2ͷ࡞Γํ 1 1 √ 2

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√ 5ͷ࡞Γํ 2 √ 5 3

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ax2 + bx + c = 0 ͷղ͸ x = −b ± √ b2 − 4ac 2a a, b, c͕࡞ਤͰ͖͍ͯΕ͹ɺ͜Ε΋Ͱ͖Δ

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ೋͭ߹Θͤͯೋ࣍ํఔࣜɻ A = (0, 1), B = (a, b)Λ௚ܘͷ྆୺ͱ͢Δԁͷࣜ x(x − a) + (y − 1)(y − b) = 0 ͜Εͱy = 0ͷަ఺͸ɺೋ࣍ํఔࣜ x2 − ax + b = 0 ͷղ

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x2 + x − 1 = 0ͷղ x = −1 ± √ 5 2 O(0, 0) A(0, 1) B(−1, −1)

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࡞ਤͰͰ͖Δ͜ͱ • ਨ௚ͳઢΛҾ͘ • ฏߦͳઢΛҾ͘ • ௕͞Λೋ౳෼͢Δ • ௕͞ͷ଍͠ࢉͱҾ͖ࢉΛ͢Δ • ௕͞ͷֻ͚ࢉͱׂΓࢉΛ͢Δ • ೋ࣍ํఔࣜΛղ͘

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1 + √ 5, 1 + 1 + √ 5, . . . ͷΑ͏ͳ௕͞ͷઢ෼΋࡞ਤͰ͖Δ

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ਖ਼5֯ܗͷ࡞ਤ

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72◦

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72◦

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72◦ 36◦ 72◦

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36◦ 36◦ 36◦ 72◦

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O A B C D 36◦ 36◦ 36◦ 72◦

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૬ࣅͳͷͰ OA : OC = OC : CD ͕ͨͬͯ͠OC = xͱͯ͠ 1 : x = x : 1 − x Ͱ͋Δɻ x2 + x − 1 = 0 ͷղΛ࡞ਤ͢Ε͹Α͍ɻ

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ࡾ֯ؔ਺Λ࢖ͬͯߟ͑Δɻ θ cos θ 1

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120◦ 1 − 1 2 cos 120◦ = cos 240◦ = − 1 2

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cos 120◦ + cos 240◦ = −1

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ࡾ֯ؔ਺ͷੑ࣭ cos(θ) = cos(360◦ − θ) cos(α + β) + cos(α − β) = 2 cos α cos β

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cos 72◦ + cos 144◦ = 2 cos 72◦ cos 144◦ 2 cos2 72◦ − 1 = cos 144◦ Ͱ͋Δ͔Β cos 72◦+2 cos2 72◦−1 = 2 cos 72◦(2 cos2 72◦−1)

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2 cos 72◦ = xͱͯ͠ 1 2 x + 2 4 x2 − 1 = x( 2 4 x2 − 1) 1 2 x3 − 1 2 x2 − 3 2 x + 1 = 0 x3 − x2 − 3x + 2 = 0 (x − 1)(x2 + x − 1) = 0 Ͱ͋Δɻ

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x2 + x − 1 = 0 ͷղx = 2 cos 72◦, 2 cos 144◦ x = −1 ± √ 5 2 2 cos 72◦+2 cos 144◦ = −1 + √ 5 2 + −1 − √ 5 2 = −1

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ղͱ܎਺ͷؔ܎ 2 cos 72◦ + 2 cos 144◦ = −1 cos 72◦ + cos 72◦ + cos 144◦ + cos 144◦ = −1 cos 72◦ + cos 144◦ + cos 216◦ + cos 288◦ = −1

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ࡾ֯ؔ਺ͷੑ࣭ Ұൠʹ cos 360◦ N +cos 2 360◦ N +· · ·+cos(N−1) 360◦ N = −1 ҰൠʹN Λح਺ͱͯ͠ cos 360◦ N +cos 2 360◦ N +· · ·+cos N − 1 2 360◦ N = − 1 2

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ೋ࣍ํఔࣜͷղͱ܎਺ͷؔ܎ x = α, β Λղʹ΋ͭೋ࣍ํఔࣜ (x − α)(x − β) = 0 x2 − (α + β)x + αβ = 0

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x2 − ax + b = 0 ͷղ͕x = α, β ͳΒ a = α + β, b = αβ

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ਖ਼17֯ܗͷ࡞ਤ

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α = cos 1 · 360◦ 17 + cos 2 · 360◦ 17 + cos 4 · 360◦ 17 + cos 8 · 360◦ 17 β = cos 3 · 360◦ 17 + cos 5 · 360◦ 17 + cos 6 · 360◦ 17 + cos 7 · 360◦ 17

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α + β = cos 1 · 360◦ 17 + cos 2 · 360◦ 17 + cos 3 · 360◦ 17 + cos 4 · 360◦ 17 + cos 5 · 360◦ 17 + cos 6 · 360◦ 17 + cos 7 · 360◦ 17 + cos 8 · 360◦ 17 = − 1 2

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αβ = cos 1 · 360◦ 17 cos 3 · 360◦ 17 + cos 1 · 360◦ 17 cos 5 · 360◦ 17 + cos 1 · 360◦ 17 cos 6 · 360◦ 17 + cos 1 · 360◦ 17 cos 7 · 360◦ 17 + cos 2 · 360◦ 17 cos 3 · 360◦ 17 + cos 2 · 360◦ 17 cos 5 · 360◦ 17 + cos 2 · 360◦ 17 cos 6 · 360◦ 17 + cos 2 · 360◦ 17 cos 7 · 360◦ 17 + cos 4 · 360◦ 17 cos 3 · 360◦ 17 + cos 4 · 360◦ 17 cos 5 · 360◦ 17 + cos 4 · 360◦ 17 cos 6 · 360◦ 17 + cos 4 · 360◦ 17 cos 7 · 360◦ 17 + cos 8 · 360◦ 17 cos 3 · 360◦ 17 + cos 8 · 360◦ 17 cos 5 · 360◦ 17

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ࡾ֯ؔ਺ͷੑ࣭ cos(θ) = cos(360◦ − θ) cos(α + β) + cos(α − β) = 2 cos α cos β θ = 360◦ 17 ͱ͢Δɻ 17θ = 360◦

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cos θ = cos 16θ cos 2θ = cos 15θ cos 3θ = cos 14θ cos 4θ = cos 13θ cos 5θ = cos 12θ cos 6θ = cos 11θ cos 7θ = cos 10θ cos 8θ = cos 9θ

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ab = cos (1θ) cos (3θ) + cos (1θ) cos (5θ) + cos (1θ) cos (6θ) + cos (1θ) cos (7θ) + cos (2θ) cos (3θ) + cos (2θ) cos (5θ) + cos (2θ) cos (6θ) + cos (2θ) cos (7θ) + cos (4θ) cos (3θ) + cos (4θ) cos (5θ) + cos (4θ) cos (6θ) + cos (4θ) cos (7θ) + cos (8θ) cos (3θ) + cos (8θ) cos (5θ) + cos (8θ) cos (6θ) + cos (8θ) cos (7θ)

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= 1 2 {cos (4θ) + cos (2θ) + cos (6θ) + cos (4θ) + cos (7θ) + cos (5θ) + cos (8θ) + cos (6θ) + cos (1θ) + cos (5θ) + cos (3θ) + cos (7θ) + cos (8θ) + cos (4θ) + cos (9θ) + cos (5θ) + cos (1θ) + cos (7θ) + cos (1θ) + cos (9θ) + cos (2θ) + cos (10θ) + cos (3θ) + cos (11θ) + cos (5θ) + cos (11θ) + cos (3θ) + cos (13θ) + cos (2θ) + cos (14θ) + cos (1θ) + cos (15θ)}

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= 1 2 {cos (4θ) + cos (2θ) + cos (6θ) + cos (4θ) + cos (7θ) + cos (5θ) + cos (8θ) + cos (6θ) + cos (1θ) + cos (5θ) + cos (3θ) + cos (7θ) + cos (8θ) + cos (4θ) + cos (8θ) + cos (5θ) + cos (1θ) + cos (7θ) + cos (1θ) + cos (8θ) + cos (2θ) + cos (7θ) + cos (3θ) + cos (6θ) + cos (5θ) + cos (6θ) + cos (3θ) + cos (4θ) + cos (2θ) + cos (3θ) + cos (1θ) + cos (2θ)}

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= 1 2 (4 cos (1θ) + 4 cos (2θ) + 4 cos (3θ) + 4 cos (4θ) + 4 cos (5θ) + 4 cos (6θ) + 4 cos (7θ) + 4 cos (8θ)) = −1 α + β = − 1 2 αβ = −1

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α, β ͸ x2 + 1 2 x − 1 = 0 ͷղͰ͋Δɻ ίϯύεͱఆنͰαͷ௕͞ͷઢ෼ͱβ ͷ௕͞ͷઢ ෼Λ࡞ਤͰ͖Δɻ

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࣍ʹ γ = cos 1 · 360◦ 17 + cos 4 · 360◦ 17 = cos (1θ) + cos (4θ) δ = cos 2 · 360◦ 17 + cos 8 · 360◦ 17 = cos (2θ) + cos (8θ) ʹ͍ͭͯγ + δ, γδΛܭࢉͯ͠ΈΔɻ

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γ + δ = cos (1θ) + cos (4θ) + cos (2θ) + cos (8θ) = α

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γδ = (cos (1θ) + cos (4θ))(cos (2θ) + cos (3θ)) = cos (1θ) cos (2θ) + cos (4θ) cos (2θ) + cos (1θ) cos (3θ) + cos (4θ) cos (3θ) = 1 2 (cos (3θ) + cos (1θ) + cos (2θ) + cos (6θ) + cos (7θ) + cos (9θ) + cos (4θ) + cos (12θ)) = 1 2 (cos (3θ) + cos (1θ) + cos (2θ) + cos (6θ) + cos (7θ) + cos (8θ) + cos (4θ) + cos (5θ))

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γ + δ = α γδ = − 1 4 γ, δ͸ x2 − αx − 1 4 = 0 ͷղͰ͋Δɻ

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ίϯύεͱఆنͰ γ = cos 1 · 360◦ 17 + cos 4 · 360◦ 17 δ = cos 2 · 360◦ 17 + cos 8 · 360◦ 17 ͷ௕͞ͷઢ෼Λ࡞ਤͰ͖Δɻ

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cos 1 · 360◦ 17 cos 4 · 360◦ 17 = 1 2 (cos 3 · 360◦ 17 + cos 5 · 360◦ 17 )

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࣍ʹ ϵ = cos 3 · 360◦ 17 + cos 5 · 360◦ 17 = cos (3θ) + cos (5θ) ζ = cos 6 · 360◦ 17 + cos 7 · 360◦ 17 = cos (6θ) + cos (7θ) ʹ͍ͭͯϵ + ζ, ϵζ Λܭࢉͯ͠ΈΔɻ

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ϵ + ζ = cos (3θ) + cos (5θ) + cos (6θ) + cos (7θ) = β

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ϵζ = (cos (3θ) + cos (5θ))(cos (6θ) + cos (7θ)) = cos (3θ) cos (6θ) + cos (5θ) cos (6θ) + cos (3θ) cos (7θ) + cos (5θ) cos (7θ) = 1 2 (cos (9θ) + cos (3θ) + cos (11θ) + cos (1θ) + cos (10θ) + cos (4θ) + cos (12θ) + cos (2θ)) = 1 2 (cos (8θ) + cos (3θ) + cos (6θ) + cos (1θ) + cos (7θ) + cos (4θ) + cos (5θ) + cos (2θ))

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ϵ + ζ = β ϵζ = − 1 4 ϵ, ζ ͸ x2 − βx − 1 4 = 0 ͷղͰ͋Δɻ

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·ͱΊΔͱ 1. x2 + 1 2 x − 1 = 0 ͷղx = α, β Λ࡞ਤ͢Δɻ 2. x2 − αx − 1 4 = 0 ͷղx = γ, δΛ࡞ਤ͢Δɻ

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3. x2 − βx − 1 4 = 0 ͷղx = ϵ, ζ Λ࡞ਤ͢Δɻ 4. x2 − γx + 1 2 ϵ = 0 ͷղx = cos 360◦ 17 ͕࡞ਤͰ͖Δɻ

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ਖ਼7֯ܗͷ࡞ਤ

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ࡾ֯ؔ਺ͷੑ࣭ Ұൠʹ cos 360◦ N +cos 2 360◦ N +· · ·+cos(N−1) 360◦ N = −1 ҰൠʹN Λح਺ͱͯ͠ cos 360◦ N +cos 2 360◦ N +· · ·+cos N − 1 2 360◦ N = − 1 2

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cos 1 · 360◦ 7 + cos 2 · 360◦ 7 + cos 3 · 360◦ 7 = − 1 2

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ഒ֯ͷެࣜ cos 2 · 360◦ 7 = 2 cos 1 · 360◦ 7 2 − 1 3ഒ֯ͷެࣜ cos 3 · 360◦ 7 = 4 cos 1 · 360◦ 7 3 − 3 cos 1 · 360◦ 7

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cos 1 · 360◦ 7 + 2 cos 1 · 360◦ 7 2 − 1 + 4 cos 1 · 360◦ 7 3 − 3 cos 1 · 360◦ 7 = − 1 2 ΛΈͨ͢ɻ

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ͭ·Γɺcos 1 · 360◦ 7 ͸ࡾ࣍ํఔࣜ 4x3 + 2x2 − 2x − 1 2 = 0 ͷղʹͳΔɻ͜Ε͕࡞ਤͰ͖Δ͔ʁ

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΋͠ೋ࣍ํఔࣜͷղʹ΋ͳͬͨͱ͢Δɻଟ߲ࣜͷ ׂΓࢉΛ͢Δͱ 4x3+2x2−2x− 1 2 = (ax2+bx+c)(dx+e)+fx+g Ͱ͋Γɺ fx + g = 0 ΋ຬͨ͞ͳ͚Ε͹͍͚ͳ͍ɻ

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࡞ਤͰ͖ͳ͍ʂ

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ෳૉ਺Ͱߟ͑Δɻ eiθ = cos θ + i sin θ ei360◦ = cos 360◦ + i sin 360◦ = 1 eiαeiβ = eiα+iβ (eiθ)n = eniθ

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(ei 360◦ 3 )3 = (e3i 360◦ 3 ) = 1 x3 = 1 x3 − 1 = (x − 1)(x2 + x + 1) = 0

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x = ei 360◦ 3 , e2i 360◦ 3 ͸x2 + x + 1 = 0ͷղͰ͋Δ

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(ei 360◦ 5 )5 = (e5i 360◦ 5 ) = 1 x5 = 1 x5 − 1 = (x − 1)(x4 + x3 + x2 + x + 1) = 0 x = ei 360◦ 5 , e2i 360◦ 5 , e3i 360◦ 5 , e4i 360◦ 5 ͸x4 + x3 + x2 + x + 1 = 0ͷղͰ͋Δ

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(ei 360◦ 7 )7 = (e7i 360◦ 7 ) = 1 x7 = 1 x7−1 = (x−1)(x6+x5+x4+x3+x2+x+1) = 0 x =ei 360◦ 7 , e2i 360◦ 7 , e3i 360◦ 7 e4i 360◦ 7 , e5i 360◦ 7 , e6i 360◦ 7 ͸x6 + x5 + x4 + x3 + x2 + x + 1 = 0ͷղͰ͋Δ

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ಉ͡ํఔࣜͷղ͸ೖΕସ͑ͯ΋۠ผͰ͖ͳ͍ʂ

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ͲΕ͚ͩೖΕସ͕͑͋Δ͔ʁ ࢛ଇԋࢉ͸อͬͨ··ɺ਺ͷೖΕସ͑Λߟ͑Δɻ

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ೋ࣍ํఔࣜͷ৔߹ −b + √ b2 − 4ac 2 , −b − √ b2 − 4ac 2 ΛೖΕସ͑Δɻ

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x4 + x3 + x2 + x + 1 = 0 ͷղͷೖΕସ͑͸ͲΕ͚ͩ͋Δ͔ʁ

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ei 360◦ 5 → e2i 360◦ 5 ͱͨ͠ͱ͢Δɻ͢Δͱ e2i 360◦ 5 = ei 360◦ 5 · ei 360◦ 5 → e2i 360◦ 5 · e2i 360◦ 5 = e4i 360◦ 5

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e3i 360◦ 5 = ei 360◦ 5 · ei 360◦ 5 · ei 360◦ 5 → e2i 360◦ 5 · e2i 360◦ 5 · e2i 360◦ 5 = e6i 360◦ 5 = ei 360◦ 5 e4i 360◦ 5 = ei 360◦ 5 · ei 360◦ 5 · ei 360◦ 5 · ei 360◦ 5 → e2i 360◦ 5 · e2i 360◦ 5 · e2i 360◦ 5 · e2i 360◦ 5 = e8i 360◦ 5 = e3i 360◦ 5 ͱ࢒Γͷղͷߦ͖ઌ΋ࣗಈతʹܾ·ͬͯ͠·͏ɻ

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Ͱ͸ei 360◦ 5 ͷߦ͖ઌ͸Կछྨ͋Δ͔ʁ ಉ͡ํఔࣜͷղͰ͋Δ ei 360◦ 5 , e2i 360◦ 5 , e3i 360◦ 5 , e4i 360◦ 5 ͷ4छྨ ͭ·Γ x4 + x3 + x2 + x + 1 = 0 ͷղͷೖΕସ͑͸4छྨ͋Δɻ

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ei 360◦ 5 + e4i 360◦ 5 ͸͜ͷ4छྨͷೖΕସ͑ͰͲͷΑ ͏ʹมԽ͢Δ͔ʁ e2i 360◦ 5 + e3i 360◦ 5 ͱೖΕସΘΔ͔ೖΕସΘΒͳ͍ ͔ͲͪΒ͔ɻ ೋ࣍ํఔࣜͷղʹͳΔʂ

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x6 + x5 + x4 + x3 + x2 + x + 1 = 0 ͷղͷೖΕସ͑͸ͲΕ͚ͩ͋Δ͔ʁ

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ei 360◦ 7 Λe2i 360◦ 7 ͷߦ͖ઌΛܾΊΕ͹ࣗಈతʹܾ ·Δ શ෦Ͱ6௨Γ

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ei 360◦ 7 + e6i 360◦ 7 ͸ e2i 360◦ 7 + e5i 360◦ 7 , e3i 360◦ 7 + e4i 360◦ 7 ͷ͍ͣΕ͔ʹͳΔɻ ࡾ࣍ํఔࣜͷղʹͳΔʂ

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p৐ͯ͠1ʹͳΔ਺ͷํఔࣜ p − 1छྨͷղͷೖΕସ͑ p − 1ͷ໿਺kʹରԠͯ͠͏·͘ղͷ૊Έ߹Θͤ Λ࡞Δ p − 1/kݸͷ૬ํ ح਺࣍ͷํఔ͕ࣜग़ͯ͘Δͱ࡞ਤͰ͖ͳ͍ɻ

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ͭ·Γp − 1͕2Ҏ֎ͰׂΕΔͱೋ࣍ํఔࣜҎ֎ ͷํఔࣜΛղ͔ͳ͚Ε͹ͳΒͳ͍ʂ ਖ਼p֯ܗ͕࡞ਤͰ͖ΔͨΊʹ͸p − 1 = 2n ͱ͍͏ ܗʹͳΔ͜ͱ͕ඞཁɻ

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